Tension of the EG statistic and RSD data with Planck/ΛCDM and implications forweakening gravity.

F. Skara∗ and L. Perivolaropoulos†

Department of Physics, University of Ioannina, 45110 Ioannina, Greece(Dated: March 26, 2020)

The EG statistic is a powerful probe for detecting deviations from GR by combining weak lensing(WL), real-space clustering and redshift space distortion (RSD) measurements thus probing boththe lensing and the growth effective Newton constants (GL and Geff ). We construct an up todate compilation of EG statistic data including both redshift and scale dependence (EG(R, z)). Wecombine this EG data compilation with an up to date compilation of fσ8 data from RSD observationsto identify the current level of tension between the Planck/ΛCDM standard model based on generalrelativity and a general model independent redshift evolution parametrization of GL and Geff . Eachfσ8 datapoint considered has been published separately in the context of independent analyses ofdistinct galaxy samples. However, there are correlations among the datapoints considered due tooverlap of the analyzed galaxy samples. Due to these correlations the derived levels of tension ofthe best fit parameters with Planck/ΛCDM are somewhat overestimated but this is the price topay for maximizing the information encoded in the compilation considered. We find that the levelof tension increases from about 3.5σ for the fσ8 data compilation alone to about 6σ when the EG

data are also included in the analysis. The direction of the tension is the same as implied by thefσ8 RSD growth data alone (lower Ωm and/or weaker effective Newton constant at low redshifts forboth the lensing and the growth effective Newton constants (GL and Geff )). These results furtheramplify the hints for weakening modified gravity discussed in other recent analyses [1–4].

I. INTRODUCTION

The theory of general relativity (GR) and the standardΛ cold dark matter (ΛCDM) [5] cosmological model havebeen remarkably successful in explaining a wide array ofobservations [6] including the observed accelerating ex-pansion of the universe [7, 8]. Despite of its successesand simplicity, the validity of the cosmological standardmodel ΛCDM is currently under intense investigation.This is motivated by a range of profound theoretical andobservational difficulties of the model. The most impor-tant theoretical difficulties of the ΛCDM model are thefine tuning [9–11] and coincidence problems [12, 13]. Thefirst of these problems corresponds to the large discrep-ancy between observations and quantum field theoreticalpredictions on the value of the cosmological constant Λwhile the second is associated with the coincidence be-tween the observed vacuum energy density ΩΛ and thematter density Ωm which in the present epoch are of thesame order of magnitude despite of their very differentevolution during the cosmic history.

A well known observational difficulty corresponds tothe tension between the cosmic microwave background(CMB) measured value of the Hubble parameter H0

[14, 15] in the context of the ΛCDM model and the localmeasurements from supernovae [16, 17] and lensing timedelay indicators [18], with local measurements suggestinga higher value. Another observational puzzle for ΛCDMinvolves persisting indications from observational probes

∗ fskara@cc.uoi.gr† leandros@uoi.gr

measuring the growth of matter perturbations that theobserved growth is weaker than the growth predicted bythe standard Planck/ΛCDM parameter values [15]. Mod-ified gravity (MG) models constitute a prime theoreticalcandidate to explain this tension.

The combination of cosmological observational probesis a powerful tool for the identification of signatures ofMG [19–25]. Such observational probes may be dividedin two classes: geometric and dynamical (or structureformation) probes [26–29]. Geometric observations mea-sure cosmological distances using standard candles (e.g.Type Ia supernovae) and standard rulers (e.g. the hori-zon at the time of recombination probed through BaryonAcoustic Oscillations) and thus probe directly the cosmicmetric, independent of the underlying theory of gravity.Dynamical observations probe the growth rate of cosmo-logical perturbations and thus the gravitational laws andthe consistency of GR with data provided the backgroundgeometry is known.

Dynamical probes include cluster counts (CC) [29–32],weak lensing (WL) [25, 33–39] and redshift-space distor-tions (RSD) [1, 2, 40–42]. These probes are consistentwith each other pointing either to a lower value of thematter density parameter Ω0m in the context of GR orto weaker gravitational growth power than the growthindicated by GR in the context of a Planck18/ΛCDMbackground geometry at about 2 − 3σ level [1, 2, 41].Such weak growth may be quantified by the parameterσ8 which is the matter density rms fluctuations withinspheres of radius 8h−1Mpc and is determined by theamplitude of the primordial fluctuations power spectrumand by the growth rate of cosmological fluctuation.

Various possible mechanisms have been proposed to

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slow down growth at low redshifts and thus reduce theabove tension (see e.g. [4]). Such mechanisms may bedivided in two categories: non-gravitational and gravita-tional. The former includes the effects of interacting darkenergy models [43–46], dynamical dark energy models[47, 48], running vacuum models [49, 50] and the effects ofmassive neutrinos [51]. The latter includes the effects ofMG theories with a reduced (compared to GR) evolvingeffective Newton’s constant Geff at low redshifts [1, 2].

The effects of MG [52–61] models are indistinguish-able from GR at the geometric cosmological backgroundlevel [26, 62, 63]. Signatures of MG can only be obtainedby investigating the dynamics of cosmological perturba-tions [64, 65] using specific statistics obtained throughdynamical probe observables such as the two-point cor-relation of and power spectrum of the galaxy distribution,the RSD and WL. A useful bias free statistic is the fσ8

product of the rate of growth of matter density pertur-bations f times σ8 discussed in more detail in what fol-lows. An alternative observable statistic is the EG whichwas constructed to be independent of both the cluster-ing bias factor b and the parameter σ8 on linear scales.This statistic was proposed in 2007 [66] and thereafterhas been used several times to test MG theories [67, 68].The expectation value of EG is equal to the ratio of theLaplacian of the sum of the Bardeen potentials [69] Ψ(the Newtonian potential) and Φ (the spatial curvaturepotential) ∇2(Ψ + Φ) over the peculiar velocity diver-gence θ ≡ ∇ · ~υ

H(z) (where ~υ is the peculiar velocity and

H(z) is the Hubble parameter in terms of the redshift z).The EG statistic has been proposed as a model inde-

pendent test of any MG theory [70] and is constructedfrom three different probes of large scale structure (LSS):the galaxy-galaxy lensing (GGL), the galaxy clusteringand the galaxy velocity field which leads to galaxy red-shift distortions. Alternatively, EG may be constructedfrom galaxy-CMB lensing [71] instead of galaxy-galaxylensing as a more robust tracer of the lensing field athigher redshifts [72, 73].

The first probe, the GGL (a special type of WL), is theslight distortion of shapes of source galaxies in the back-ground of a lens galaxy, which arises from the gravita-tional deflection of light due to the gravitational potentialof the lens galaxy along the line of sight (see for example[74–77]). This WL probe is sensitive to ∇2(Ψ + Φ), sincerelativistic particles collect equal contributions from thetwo Bardeen potentials which appear in the scalar per-turbed Friedmann-Lemaıtre-Robertson-Walker (FLRW)metric in the Newtonian gauge [78–80]

ds2 = −(1 + 2Ψ)dt2 + a2(1− 2Φ)d~x2 (1.1)

where a is the scale factor that is related to the redshiftz through a = 1

1+z .The second probe, the galaxy clustering arises from

the gravitational attraction of matter and is sensitiveonly to the potential Ψ. Similarly, the third probe, thegalaxy velocity field, is quantified by measuring redshiftspace distortions (RSD) [81–84] (an illusory anisotropy

that distorts the distribution of galaxies in redshift spacegenerated by their peculiar motions falling towards over-dense regions). This important probe of LSS is sensitiveto the rate of growth of matter density perturbationsf which depends on the theory of gravity and providesmeasurements of fσ8 that depends on the potential Ψ.

In most MG theories the potentials Φ and Ψ obey gen-eralized Poisson equations like the GR Newtonian po-tential where the MG effects are encoded in generalizedspace-time dependent effective Newton constants. Thesegeneralized Newton constants for the potential Ψ and forthe lensing combination Ψ + Φ are usually described bytwo parameters: the effective Newton’s constant param-eter µ and the light deflection parameter Σ. In the mod-ified Poisson equations [85] the µ and Σ are connectedwith the potentials Ψ and Ψ + Φ respectively. In GRthe value of µ and Σ coincides with unity while in a MGmodel µ and Σ can be in general functions of both timeand scale [19, 86]. Using fσ8 and EG datasets constraintscan be imposed on the parameters µ and Σ [23, 87–92]).Such analyses have revealed various levels of tension ofthe best fit forms of µ and Σ with the GR predictionof unity showing hints that these parameters may be lessthan unity implying weaker growth of perturbations thanthat predicted in GR. The goal of the present analysis isto extend these studies and use an updated data compi-lation for both the fσ8 and EG statistics to identify thecurrent level of tension with GR implied by these datacompilations.

In particular, we address the following questions:

• What are efficient phenomenological redshift de-pendent parametrizations of the generalized nor-malized Newton constants µ(z) and Σ(z) that areconsistent with solar system and nucleosynthesisconstraints that indicate that GR is restored athigh z and at the present time in the solar system?

• What are the constraints imposed by the EG andfσ8 updated data compilations on the parametersof the above parametrizations and do these con-straints amplify the hints for weakening gravity atlow z implied by the fσ8 data alone as indicatedby previous studies?

The plan of this paper is the following: In the nextSection II we present a brief review of the theoreticalexpression for EG. We also present phenomenologicallymotivated parametrizations for µ and Σ and describehow we use them in order to probe possible deviationsfrom GR on cosmological scales. In Section III weuse compilations of fσ8 and EG data along with thetheoretical expressions for fσ8 and EG which involve µand Σ to derive constraints on these parameters and toidentify the tension level between the Planck/ΛCDMparameter values favoured by Planck 2018 [15] shown inTable I and the corresponding parameter values favoredby the two datasets. Finally in Section IV we conclude,summarize and discuss the implications and possible

3

TABLE I. Planck18/ΛCDM parameters values [15] based on TT,TE,EE+lowE+lensing likelihoods.

Parameter Planck18/ΛCDM

Ωbh2 0.02237± 0.00015

Ωch2 0.1200± 0.0012

nS 0.9649± 0.0042H0 [kms−1Mpc−1] 67.36± 0.54

Ω0m 0.3153± 0.0073w −1σ8 0.8111± 0.0060

extensions of our analysis.

II. THEORETICAL BACKGROUND

II.1. EG statistic

The EG statistic [66, 70] is designed as a probe of theratio of the Bardeen potentials of the perturbed FRWmetric (1.1) in such a way as to be independent of theeffects of galaxy bias at linear order. It is defined as theratio of the cross correlation power spectrum Pg∇2(Φ+Ψ)

between lensing maps (cosmic shear or CMB) and galaxypositions, over the the cross-correlation power spectrumPgθ between galaxies and velocity divergence field θ

EG ≡Pg∇2(Φ+Ψ)

Pgθ(2.1)

In Fourier space the EG statistic may also be expressedas [66]

EG(l,∆l) =Cκg(l,∆l)

3H20a−1∑α qα(l,∆l)Pαvg

(2.2)

where H0 is the Hubble parameter today, l is the mag-nitude of two-dimensional wavenumber of the on-skyFourier space, Cκg(l,∆l) is the galaxy-galaxy lensingcross correlation power spectrum in bins of ∆l, Pαvg isthe galaxy-velocity cross correlations power spectrum be-tween kα and kα+1 (where k three-dimensional wavenum-ber of the on-sky Fourier transform with k1 < k2 < ... <kα < ...) and qα(l,∆l) is the weighting function definedaccordingly.

The corresponding expectation value of EG, averagedover l is the the ratio of the Laplacian of the gravitationalscalar potentials Ψ and Φ which appear in the scalar per-turbed Friedmann-Lemaıtre-Robertson-Walker (FLRW)metric Eq. (1.1) over the peculiar velocity divergence [67]

〈EG〉 =

[∇2(Ψ + Φ)

3H20a−1θ

]k=l/χ,z

(2.3)

where χ is the comoving mean distance corresponding tothe mean redshift z.

In ΛCDM cosmology and assuming that the velocityfield is generated under linear perturbation theory, thepeculiar velocity divergence is connected to the growthrate f as θ = fδ [93] where δ ≡ δρ

ρ is the matter over-

density field, ρ is the matter density of the background,

f(a) ≡ d lnD(a)d ln a is the linear growth rate of structure and

D(a) ≡ δ(a)δ(a=1) the growth factor.

In the case of GR and in the absence of any anisotropicstress the Bardeen potentials are equal (Ψ = Φ) and thegravitational field equations reduce to Poisson equationsof the form

∇2Φ = ∇2Ψ = 4πGa2ρδ =3

2H2

0 Ω0ma−1δ (2.4)

where Ω0m = Ωm(z = 0) is the matter density parametertoday and the second equality is straightforwardly de-rived assuming non-relativistic matter species and using

the equations H20 =

8πGρc,03 , ρ = ρ0a

−3 and Ω0m = ρ0ρc,0

(with ρ0 the matter density today and ρc,0 the criticaldensity today).

Therefore within GR Eq.(2.4), the Eq.(2.3) reduce to

EG =Ω0m

f(z)(2.5)

where f is well approximated as f(z) ' Ωγm(z) with thegrowth index γ in a narrow range near 0.55, for a widevariety of dark-energy models in GR [94–102]. Note thatEG in GR is scale independent (Eq.(2.5)). This is notnecessarily the case in the context of MG theories wherethe growth rate f may be strongly scale dependent evenon subhorizon scales.

II.2. The effective Newton’s constant parameter µand the light deflection parameter Σ

The gravitational slip parameter η describes the pos-sible inequality [103, 104] of the two Bardeen potentialsthat may occur in MG theories. It is defined as

η(a, k) =Φ(a, k)

Ψ(a, k)(2.6)

4

Clearly an observation of η 6= 1 would indicate physicsbeyond GR. In this case the gravitational field equationsat linear level take the form of Poisson equations thatgeneralize Eqs. (2.4). At linear level, in MG models,using the perturbed metric (1.1) and the gravitationalfield equations the following phenomenological equationsemerge [42, 86, 105–110] for the scalar perturbation po-tentials

k2(Ψ + Φ) = −8πGNΣ(a, k)a2ρ∆ (2.7)

k2Ψ = −4πGNµ(a, k)a2ρ∆ (2.8)

where ρ is the matter density of the background, ∆ thecomoving matter density contrast defined as ∆ ≡ δ +3Ha(1 + w)υ/k which is gauge-invariant [106], w = p/ρis the equation-of-state parameter and υi = −∇iu is theirrotational component of the velocity field. Also µ and Σare the generalized growth and lensing effective Newtonconstants. They are in general functions of time and scaleencoding the possible modifications of General Relativitydefined as1

µ(a, k) ≡ Geff (a, k)

GN

Σ(a, k) ≡ GL(a, k)

GN

with GN is the Newton’s constant as measured by lo-cal experiments, Geff is the effective Newton’s constantwhich is related to the growth of matter perturbationand GL is related to the lensing of light (the propagationof relativistic particles, such as photons when they tra-verse equal regions of space and time along null geodesicsexperiencing gravitational lensing collecting equal contri-butions from two gravitational potentials Ψ and Φ). Us-ing the gravitational slip Eq.(2.6) and the ratios of the

Poisson equations (2.7), (2.8) defined above the two LSSfunctions µ and Σ are related via

Σ(a, k) =1

2µ(a, k) [1 + η(a, k)] (2.9)

In GR which predicts a constant homogeneous Geff =GN , we obtain µ = 1, η = 1 and Σ = 1.

Notice that Eqs. (2.7) and (2.8) indicate that a pos-sible observation of reduced gravitational growth of theBardeen potentials may be interpreted either as reducedstrength of gravitational interaction (reduced µ and/orΣ) or due to reduced matter density ρ (or Ω0m). In thecontext of a fixed value of matter density determined bygeometric probes of the cosmological background, the re-duced gravitational growth could be either interpretedas a tension within the ΛCDM parameter value for thematter density or as a hint for weakening gravity. Indeed,such hints of weaker than expected gravitational growthof the Bardeen potentials has been observed at low red-shifts by a wide range of dynamical probes including RSDobservations [1, 2, 41, 42], WL [25, 34, 36–39] and CCdata [29–32]. In most cases this weak growth has beeninterpreted as a tension for the parameters σ8 and Ω0m

which are found by dynamical probes to be smaller thanthe values indicated by geometric probes in the contextof ΛCDM .

The observables fσ8(a, k) and EG(a, k) can probe di-rectly the gravitational strength functions µ(a, k) andΣ(a, k). In particular fσ8 is easily expressed in termsof the amplitude σ8 and the matter overdensity δ usingthe matter overdensity evolution equation (see e.g. [80])

δ + 2Hδ − 4πGNµ(a, k)ρδ ' 0 (2.10)

where the dot denotes differentiation with respect to cos-mic time t. In terms of redshift Eq. (2.10) takes the form[1, 80]

δ′′(z) +

((H(z)2)′

2 H(z)2− 1

1 + z

)δ′(z)− 3

2

(1 + z) Ω0m µ(z, k)

H(z)2/H20

δ(z) = 0 (2.11)

where primes denote differentiation with respect to the redshift. While in terms of the scale factor we have [52, 101, 111]

δ′′(a) +

(3

a+H ′(a)

H(a)

)δ′(a)− 3

2

Ω0mµ(a, k)

a5H(a)2/H20

δ(a) = 0 (2.12)

here primes denote differentiation with respect to thescale factor. In Eqs. (2.11), (2.12) possible devia-tions from GR are expressed by allowing for a scale and

1 Note that, in the literature µ and Σ are also referred to as GM

and GL (e.g. in Refs. [1, 109]) or as Gmatter and Glight (e.g. inRefs. [42, 107]).

redshift-dependent µ = µ(z, k). In the present sectionand in section III.1 we ignore scale dependence due to thelack of good quality scale dependent fσ8 and EG data.However, in section III.2 we discuss the scale dependenceof EG data.

For a given parametrization of µ(a) and initial condi-tions deep in the matter era where GR is assumed to bevalid leading to δ ∼ a equations (2.11), (2.12) may be

5

easily solved numerically leading to a predicted form ofδ(a) for a given Ω0m and background expansion H(z). Inthe context of the present analysis we assume a ΛCDMbackgroung H(z)

H2(z) = H20

[Ω0m(1 + z)3 + (1− Ω0m)

](2.13)

Once the evolution of δ is known, the observable productfσ8(a) ≡ f(a) ·σ(a) can be obtained using the definitions

f(a) ≡ d ln δ(a)

d ln a(2.14)

σ(a) ≡ σ8δ(a)

δ(a = 1)(2.15)

where σ(a) is the redshift dependent rms fluctuationsof the linear density field within spheres of radius R =8h−1Mpc and σ8 is its value today. Thus, we have

f σ8(a, σ8,Ω0m, µ) =σ8

δ(a = 1)a δ′(a,Ω0m, µ) (2.16)

This theoretical prediction may now be used to comparewith the observed fσ8 data and obtain fits for the param-eters Ω0m, σ8 and µ(z) (assuming a specific parametriza-tion of µ(z)).

The lensing gravity parameter Σ(z) can be fit in thecontext of specific parametrizations using its connectionwith the EG(a) observable as [112–114]

EG(a,Ω0m, µ,Σ) =Ω0mΣ(a)

f(a,Ω0m, µ)(2.17)

This equation assumes that the redshift of the lens galax-ies can be approximated by a single value while EG cor-responds to average value along the line of sight [114].In the context of Eq. (2.17) and assuming a specificparametrization for µ and Σ, the theoretical predictionfor EG may be used to compare with the observed EGdatapoints and lead to constraints on Ω0m, µ,Σ. Theseconstraints may be considered either separately fromthose of the fσ8 data or jointly by combining the EG andfσ8 datasets. The allowed range of these parameters maythen be compared with the standard Planck/ΛCDM pa-rameter values µ = 1, Σ = 1, Ω0m = 0.315±0.0073, σ8 =0.811± 0.006 to identify the likelihood of Planck/ΛCDMin the context of the dynamical probe data EG and fσ8

. This plan is implemented in what follows in the con-text of specific parametrizations describing the possibleevolution of µ and Σ.

On scales much smaller than the Hubble scale for mostmodified gravity models the scale dependence of µ andΣ is weak. For example in scalar-tensor (ST) model (fork aH) µ is independent of the scale [115]. Thus, westart by considering scale independent parametrizationsfor µ and Σ which reduce to the GR value at early timesand at the present time as indicated by solar system (ig-noring possible screeing effects) and Big Bang Nucleosyn-thesis constraints (µ = 1 and µ′ = 0 for a = 1 and µ = 1for a 1) [116–118]. Such parametrizations are of theform [1, 2, 119]

µ = 1 + ga(1− a)n − ga(1− a)2n = 1 + ga(z

1 + z)n − ga(

z

1 + z)2n (2.18)

Σ = 1 + gb(1− a)m − gb(1− a)2m = 1 + gb(z

1 + z)m − gb(

z

1 + z)2m (2.19)

where ga and gb are parameters to be fit and n and mare integer parameters with n ≥ 2 and m ≥ 2 which weset equal to 2 in the present analysis.

III. OBSERVATIONAL CONSTRAINTS

III.1. Scale Independent Analysis

The fσ8(z) and EG(z) updated data compilations usedin our analysis are shown in Tables VI and VII of the Ap-pendix B along with the references where each datapointwas originally published. The datapoints are also shownin Figs. 1 and 2 along with curves corresponding to thePlanck/ΛCDM prediction and the best fit parameter val-

ues. As it can be seen the datapoints from the varioussurveys are consistent with each other at any given red-shift and at 1σ level. Clearly, in both cases the data ap-pear to favor lower values of fσ8 and EG than the valuescorresponding to the Planck/ΛCDM parameters. Thistrend combined with the indications for a Planck/ΛCDMbackground from geometric probes may be interpreted asa need for a new degree of freedom which in our approachis coming from MG. In addition, we see that there is notension between different fσ8 datapoints. Instead, thereis a combined trend of the datapoints to be in tensionwith the Planck/ΛCDM prediction. This tension disap-pears when we keep the same ΛCDM background butallow for a MG evolution of the effective Newton’s con-stant. In fact, this trend may be shown to be translated

6

GR-ΛCDM Planck18

Best Fit Ωm = 0.272, ga=-1.306, σ8=0.886 , n=2)

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

z

fσ8(z)

FIG. 1. The fσ8(z) data compilation from Table VI used inthe present analysis. The subset of the data with less corre-lation is indicated with dark red. The red curve shows thePlanck18/ΛCDM prediction (parameter values Ω0m = 0.315,ga = 0, σ8 = 0.811), the blue curve shows the best fitof the fσ8(z) in the context of parametrizations Eq.(2.18)with a ΛCDM background (parameter values Ω0m = 0.272,ga = −1.306, σ8 = 0.886) and the shaded regions correspondto 1σ confidence level around the best fit (see also Table II).

GR-ΛCDM Planck18

Best Fit (Ω0 m=0.313, ga=-0.129, gb=-2.308 , n=2, m=2)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

z

EG(z)

FIG. 2. The EG(z) data compilation from Table VII (scales3 < R < 150h−1Mpc) used in the present analysis. The sub-set of the data with less correlation is indicated with darkred. The red curve shows the theoretical prediction basedon the Planck18/ΛCDM parameter values (Ω0m = 0.315,σ8 = 0.811, µ = 1, Σ = 1), the blue curve shows the bestfit theoretical prediction based on the parametrizations (2.18)and (2.19) with parameter values (Ω0m = 0.313, ga = −0.129,gb = −2.308). Notice that the best fit is significantly belowthe Planck/ΛCDM theoretical prediction and implies weakergravity (µ < 1 and Σ < 1) at the 4.6σ level (see also TableII).

into a trend for lower values for the gravitational pa-rameters µ and Σ and is quantified through a detailedmaximum likelihood analysis.

Each fσ8(z) and EG(z) datapoint of the compilationsof Tables VI and VII has been published separately in thecontext of independent analyses of distinct galaxy sam-ples and lensing data. However, the correlations amongthe datapoints considered due to overlap of the analyzed

galaxy samples may lead to an amplification of the ex-isting trends indicated by the data and an amplificationof the existing tension of the best fit parameters withPlanck/ΛCDM . Despite of this fact we have chosen tokeep the relatively large number of distinct publisheddatapoints in order to maximize the information encodedin the compilations considered keeping in mind that thismay lead to an artificial amplification of the trends thatalready exist in the data.

An additional motivation for keeping the full set ofpublished datapoints is that it is not always clear whichone of the correlated points is more suitable to keep. Ig-noring one of the correlated points arbitrarily or simplybased on time of publication criteria could lead to loss ofuseful information or selection bias.

Keeping the full set of points does not significantlychange the results and the level of tension between thegrowth data best fit parameter values corresponding toMG and Planck/ΛCDM best fit in the context of GR. Inorder to demonstrate the validity of the above reasonswe have repeated our analysis for a subset of the fσ8

and EG data where we have removed most earlier datathat were subject to correlations with more recent dataas indicated with bold font in the index of the Tables VIand VII and as shown in Figs. 1 and 2 with dark red.The result was a data compilation of about half the fσ8and EG datapoints with significantly less correlation.The results of the statistical analysis of this dataset arepresented in Appendix A and indicate a minor reductionof the overall tension.

For the construction of the likelihood contours of themodel parameters in the context of the fσ8 and EGdatasets we construct χ2

fσ8and χ2

EGFor the construc-

tion of χ2fσ8

we use the vector [2]

V ifσ8(zi, p) ≡ fσobs8,i −

fσth8 (zi, p)

q(zi,Ω0m,Ωfid0m)

(3.1)

where fσobs8,i is the the value of the ith datapoint, withi = 1, ..., Nfσ8

(where Nfσ8= 66 corresponds to the total

number of datapoints of Table VI) and fσth8 (zi, p) is thetheoretical prediction, both at redshift zi. The parame-ter vector p corresponds to the parameters σ8,Ω0m, ga ofEq. (2.16) with the parametrization (2.18). The fiducialAlcock-Paczynsk correction factor q [1, 2, 41] is definedas

q(zi,Ω0m,Ωfid0m) =

H(zi)dA(zi)

Hfid(zi)dfidA (zi)

(3.2)

where H(z), dA(z) correspond to the Hubble parameterand the angular diameter distance of the true cosmologyand the superscript fid indicates the fiducial cosmologyused in each survey to convert angles and redshift to dis-tances for evaluating the correlation function. As shownin Table II, the effects of this correction factor are lessthan about 10% in the derived best fit parameter values.

7

Best fit

fσ8 data

Planck ΛCDM

0.1 0.2 0.3 0.4 0.5 0.60.6

0.7

0.8

0.9

1.0

1.1

1.2

Ω0m

σ 8 Best fit

fσ8 data

Planck ΛCDM

0.1 0.2 0.3 0.4 0.5 0.6-3

-2

-1

0

1

2

Ω0m

ga

Best fit

fσ8 data

Planck ΛCDM

-4 -2 0 2 40.4

0.6

0.8

1.0

1.2

1.4

ga

σ 8

FIG. 3. The three 1σ - 7σ confidence contours in 2D projected parameter spaces of the parameter space (Ω0m, σ8, ga) in thecontext of parametrization Eq.(2.18) with n = 2 including the fiducial correction factor Eq. (3.2). The RSD data fσ8(z) fromTable VI of the Appendix B was used. The third parameter in each contour was fixed to the best fit value. The red and greendots describe the Planck18/ΛCDM best fit and the best-fit values from data.

Best fit

EG (z) data

Planck ΛCDM

0.0 0.1 0.2 0.3 0.4 0.5-6

-4

-2

0

2

Ω0m

ga

Best fit

EG (z) data

Planck ΛCDM

-4 -2 0 2 4-8

-6

-4

-2

0

2

4

ga

gb

Best fit

EG (z) data

Planck ΛCDM

0.0 0.2 0.4 0.6 0.8-8

-6

-4

-2

0

2

4

6

Ω0m

gb

FIG. 4. The three 1σ - 5σ confidence contours in 2D projected parameter spaces of the parameter space (Ω0m, ga, gb) in thecontext of parametrizations Eqs.(2.18) and (2.19) with n = 2,m = 2. The data EG(z) from Table VII of the Appendix B wereused. The third parameter in each contour was fixed to the best fit value. The red and green dots describe the Planck18/ΛCDMbest fit and the best-fit values from data.

Thus we obtain χ2fσ8

as

χ2fσ8

(Ω0m, σ8, ga) = V ifσ8Ffσ8,ijV

jfσ8

(3.3)

where Ffσ8,ij is the Fisher matrix (the inverse of thecovariance matrix Cfσ8,ij of the data) which is assumedto be diagonal with the exception of the 3 × 3 WiggleZsubspace (see [2] for more details on this compilation).

Similarly, for the construction of χ2EG

, we consider thevector

V iEG(zi, p) ≡ EobsG,i − EthG (zi, p) (3.4)

where EobsG,i is the the value of the ith datapoint, with

i = 1, ..., NEG(where NEG

= 16 corresponds to the totalnumber of datapoints of Table VII), while EthG (zi, p) is thetheoretical prediction (Eq. (2.17)), both at redshift zi.

The parameter vector p corresponds to the parameters ofEq. (2.17) with the parametrization (2.18) namely Ω0m,ga, gb.

Thus we obtain χ2EG

as

χ2EG

(Ω0m, ga, gb) = V iEGFEG,ijV

jEG

(3.5)

where FEG,ij is the Fisher matrix also assumed to bediagonal.

By minimizing χ2fσ8

, χ2EG

separately and combined

as χ2tot ≡ χ2

fσ8+ χ2

EGwe obtain the constraints on the

parameters Ω0m, σ8, ga, gb shown in Figs. 3, 4 and5 respectively. Each one of these Figures correspondsto a 2D projection that goes through the best fitparameter point in parameter space of the full threeor four dimensional contour plot in each case. Thefull number of parameters (three or four) was assumed

8

Best fit

EG+ fσ8 data

Planck ΛCDM

0.1 0.2 0.3 0.4 0.50.7

0.8

0.9

1.0

1.1

Ω0m

σ 8

Best fit

EG+ fσ8 data

Planck ΛCDM

0.1 0.2 0.3 0.4 0.5-2.0

-1.5

-1.0

-0.5

0.0

0.5

Ω0m

ga

Best fit

EG+ fσ8 data

Planck ΛCDM

-3 -2 -1 0 1 20.6

0.7

0.8

0.9

1.0

1.1

1.2

ga

σ 8

Best fit

EG+ fσ8 data

Planck ΛCDM

-3 -2 -1 0 1 2-6

-5

-4

-3

-2

-1

0

1

ga

gb Best fit

EG+ fσ8 data

Planck ΛCDM

0.1 0.2 0.3 0.4 0.5-6

-4

-2

0

2

Ω0m

gb

Best fit

EG+ fσ8 data

Planck ΛCDM

-6 -4 -2 0 20.70

0.75

0.80

0.85

0.90

0.95

1.00

gb

σ 8

FIG. 5. The six 1σ - 7σ confidence contours in 2D projected parameter spaces of the parameter space (Ω0m, σ8, ga, gb) in thecontext of parametrizations Eqs.(2.18)and (2.19) with n = 2 and m = 2 including the fiducial correction factor Eq. (3.2).The data EG(z) and fσ8(z) from Tables VII and VI of the Appendix B was used. The third and the forth parameter in eachcontour were fixed to the best fit values. The red and green dots describe the Planck18/ΛCDM best fit and the best-fit valuesfrom data.

when constructing the contour 2D projections. Previousstudies [1, 2] have considered similar 2D projections thatgo through the Planck/ΛCDM best fit parameter pointin the higher dimensional parameter space. This laterchoice tends to change somewhat (in most projections itis increased) the apparent tension between the best fitMG parameter values and the best fit Planck/ΛCDMparameters in the 2D projection parameter subspaces.This 2D tension may be in some cases misleading due toprojection effects and thus in Table (III) we stress thetension in the full 3D or 4D parameter space.

The tension level between the best fit MG parameter

values and the Planck/ΛCDM best fit parameter valuesis significant in both the 2D projection parameter spacesshown in Figs 3, 4 and 5 and in the higher 3D parameterspace likelihood surfaces shown in Fig. 6. The bestfit parameter values obtained in the context of thedatasets considered and the tension levels in both the2D projections and in the full 3D-4D parameter spacesare shown in Tables II and III respectively. In theseTables we also show the cases corresponding to fitswithout including the correction factor (3.2) in the fσ8

data demonstrating that there is a small change in thebest fit parameter values.

The following comments can be made on the resultsshown in Figs. 3, 4 and 5 and Tables II and III:

• The left part of Table III shows the tension levelin the full 3D or 4D parameter space. The ten-

sion level between Planck/ΛCDM and best fit MGmodel parametrizations (2.18) and (2.19) in thecontext of the fσ8 data is significant (about 3.5σ)but is is less than the corresponding tension ob-

9

FIG. 6. Left: The 1σ - 2σ confidence contour of the parameter space (Ω0m, σ8, ga) in the context of parametrization Eq.(2.18)with n = 2 including the fiducial correction factor Eq. (3.2). The RSD data fσ8(z) from Table VI of the Appendix B was used.The red and green dots describe the Planck18/ΛCDM best fit and the best-fit values from data. Right: The 1σ - 2σ confidencecontour of the parameter space (Ω0m, ga, gb) in the context of parametrizations Eqs.(2.18) and (2.19) with n = 2. The dataEG(z) from Table VII of the Appendix B was used. The red and green dots describe the Planck18/ΛCDM best fit and thebest-fit values from data. The 3D contours include only the surfaces in 3D while the intermediate space is not filled. Thus, thewhite gaps that appear in the right figure between the surfaces, simply correspond to the white background seen from behind.

Param. Planck18/ΛCDM Dataset Dataset Dataset Datasets Datasetsfσ8(z). fσ8(z) EG(z) fσ8(z) + EG(z) fσ8(z) + EG(z)

corr. no corr. corr. no corr.

Ω0m 0.3153± 0.0073 0.272± 0.019 0.263± 0.015 0.313± 0.024 0.275± 0.015 0.264± 0.012σ8 0.8111± 0.0060 0.886± 0.015 0.90± 0.016 0.848± 0.015 0.879± 0.015ga 0 −1.306± 0.140 −1.331± 0.138 −0.129± 0.490 −0.957± 0.144 −1.115± 0.137gb 0 −2.308± 0.423 −2.448± 0.414 −2.422± 0.416

TABLE II. Planck18/ΛCDM based on TT,TE,EE+lowE+ lensing likelihoods best fit [15] and the best-fit values from data.

tained using the EG statistic data (more than 4σ).In fact for the combined fσ8 + Eg dataset the ten-sion level increases to close to 6σ! This significanttension level comes independently from both thefσ8 and EG data and hints towards weaker gravity(µ and Σ lower than 1) compared to the predic-tions of GR at low z. We stress however that thisextreme level of tension is partly due to correlationsamong the considered datapoints which necessarilyexist in our compilations.

• The weaker than expected gravitational growth in-dicated by the data is expressed as both a lowerbest fit Ω0m than expected from ΛCDM and as neg-ative best fit values for the gravitational strengthevolution parameters ga and gb (see e.g. Fig. 5).

• Ignoring the fiducial model correction factor of Eq.(3.2) in most cases tends to slightly increase thetension level (compare e.g. the last two lines ofTable III). Thus the consideration of this correctionin our analysis is a conservative approach.

The trend for weaker gravity at low redshifts is alsoevident in Fig. 7 which shows the best fit form of µ(a)

and Σ(a) in the context of each dataset.

Also, the likelihood contours in the σ8-Ωm parameter

10

Space 2D Projected SpaceDataset (Ω0m, σ8, ga) (Ω0m, ga, gb) (Ω0m, σ8, ga, gb) (Ω0m, σ8) (Ω0m, ga) (σ8, ga) (ga, gb) (Ω0m, gb) (σ8, gb)

fσ8(z) corr. 3.70σ 3.00σ ∼ 8σ 2.08σfσ8(z) no corr. 4.15σ 2.75σ ∼ 8σ 1.13σ

EG(z) 4.57 0.002σ 4.45σ 4.94σEG(z)+fσ8(z) corr. 6.03 1.47σ 6.39σ 2.59σ 5.74σ 7.74σ 5.58σEG(z)+fσ8(z)no corr. 6.33 2.17σ ∼ 8σ 2.16σ 7.53σ ∼ 8σ 6.74σ

TABLE III. Sigma differences of the best fit contours from Planck18/ΛCDM.

GR-ΛCDM

fσ8 data

Eg data

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

a

μ(a)

GR-ΛCDM

Eg data

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

a

Σ(a)

FIG. 7. Evolution of µ and Σ as functions of the scale factor a considering the best fit values for ga and gb in the contextof parametrizations Eqs.(2.18) and (2.19) with n = 2,m = 2. The data EG(z) and fσ8(z) from Tables VII and VI of theAppendix B was used. The dashed curves correspond to 1σ deviations of the parameters µ and Σ. The red lines correspond tothe GR-ΛCDM model.

space obtained using the growth data in the presenceof the MG parameter ga and in the context of GR(ga = 0) are shown in Fig. 8. We have considered boththe case of a marginalized MG parameter value andthe case of setting ga to its best fit value. Clearly thetension level between the best fit parameter values andPlanck/ΛCDM decreases significantly in the presence ofthe MG parameter ga.

The introduction of additional parameters of anytype would in general widen the likelihood contoursand thus reduce the tension between growth data andgeometric/CMB data. In general a faster expansionrate (w < −1) would tend to reduce the growth rate ofperturbations in agreement with dynamical observables.However, geometric observables (SnIa, BAO etc.) donot allow significant deviations of the expansion ratefrom ΛCDM. Thus the most efficient way to produce aweaker growth of perturbations is the introduction ofevolution of the MG parameters µ and Σ. In Fig 9 wehave demonstrated this effect by fixing ga = 0, gb = 0and constructing the σ8-Ωm contours with w = −1 andw free to vary in a range ([−1.5,−0.5]) consistent withgeometric probes. The reduction of the tension in thiscase is significantly smaller compared to the introductionof MG degrees of freedom.

III.2. Scale Dependent Data Compilations

Scale dependent parametrizations for µ and η can de-scribe a large class of MG models [86, 106]. For examplea scale dependent class of parametrizations predicted byscalar-tensor theories for µ and η is of the form [120, 121]

µ(a, k) = 1 + f1(a)1 + c1(λH/k)2

1 + (λH/k)2(3.6)

η(a, k) = 1 + f2(a)1 + c2(λH/k)2

1 + (λH/)2(3.7)

where f1 and f2 are properly chosen functions that de-pend on the scale factor. Thus a physically motivatedscale dependent generalization of the parametrizations(2.18) and (2.19) for µ and Σ may be written as

µ(R, z) = 1+

[ga(

z

1 + z)n − ga(

z

1 + z)2n

]1 + sa(λHR)2

1 + (λHR)2

(3.8)

Σ(R, z) = 1+

[gb(

z

1 + z)m − gb(

z

1 + z)2m

]1 + sb(λHR)2

1 + (λHR)2

(3.9)

11

Best fit

fσ8 data

ga=0

Planck ΛCDM

0.0 0.1 0.2 0.3 0.4 0.5 0.60.6

0.7

0.8

0.9

1.0

1.1

Ω0m

σ 8

Best fit

fσ8 data

ga to best fit value

Planck ΛCDM

0.0 0.1 0.2 0.3 0.4 0.5 0.60.6

0.7

0.8

0.9

1.0

1.1

Ω0m

σ 8

Best fit

fσ8 data

marginalization over ga

Planck ΛCDM

0.0 0.1 0.2 0.3 0.4 0.5 0.60.6

0.7

0.8

0.9

1.0

1.1

Ω0m

σ 8

FIG. 8. The confidence contours of the parameter space (σ8-Ωm) in the context of GR (left panel) and in the presence of theMG parameter ga (fixing w = −1) in the context of parametrization Eq.(2.18) with n = 2. We have considered both the caseof a marginalized MG parameter value (right panel) and the case of setting ga to its best fit value (middle panel). The red andgreen dots describe the Planck18/ΛCDM best fit and the best-fit values from data. The fσ8(z) data compilations of datapointswith less correlation from Table VI of the Appendix B was used. Notice the reduction of tension between the growth data bestfit and the Planck/ΛCDM parameter values when the MG degree of freedom is introduced.

where sa, sb and λ are parameters to be determined froma proper scale dependent dataset. Such a scale depen-dent data compilation for the statistic EG in two redshiftranges is shown in Fig. 10 and in Tables VIII and IX forlow and high z respectively in the Appendix B. The anal-ysis of this compilation may be performed in the contextof the scale dependent parametrizations (3.8) and (3.9).Clearly as shown in Fig. 10, for both low and high zthe scale independent MG parametrizations of Eqs.(2.18)and (2.19) at z = 0.3 and at z = 0.7, lead to a best fitvalue of EG that is lower compared to the Planck/ΛCDMprediction. The full scale dependent analysis leads tosimilar levels of tension as those indicated in Table IIIfor the scale independent case and will be presented indetail elsewhere.

IV. CONCLUSIONS-DISCUSSION

We have used up to date compilations of EG and fσ8

data (Tables VI and VII) based on WL and RSD ob-servations to obtain updated estimates of the tensionbetween the Planck/ΛCDM best fit parameter valuesand the best fit parameter values obtained in the con-text of an effective MG gravity model allowing for prop-erly parametrized evolution of the growth and lensinggravitational constants µ and Σ. The scale independentparametrizations (Eqs.(2.18), (2.19)) of µ and Σ dependon the parameters ga and gb respectively and are byconstruction consistent with GR at early times and atpresent as indicated by nucleosynthesis and solar systemconstraints assuming no screening is present. We have as-sumed a flat ΛCDM expansion background and we thusfit the parameters (Ω0m, σ8, ga, gb).

We find that the EG data amplify the previouslywell known indications for low Ω0m and/or weakergravity (µ < 1) at low z and favor weaker grav-ity for both the growth and the lensing gravita-tional constants (µ < 1 and Σ < 1). The ten-sion level between the Planck/ΛCDM parameter values(Ω0m, σ8, ga, gb) = (0.31, 0.81, 0, 0) and the best fit pa-rameter values obtained using the combined EG + fσ8

dataset (Ω0m, σ8, ga, gb) = (0.28, 0.85,−0.96,−2.45) is 6σwhich is significantly larger compared to the tension ob-tained when only the fσ8 dataset is used (3.7σ as shownin Table III). Even though the absolute magnitude of thederived tension is overestimated due to the correlationsamong the datapoints the amplified trend for weakergravity at low z is clearly indicated by both the fσ8 andEG data compilations and appears to be stronger for thecase of the EG data.

If this trend has some physical origin and is not dueonly to data systematics or physical effects in the con-text of GR, there are significant implications for theo-retical models. In particular f(R) theories genericallypredict stronger gravity at low z compared to its presenttime[122] (thus the prediction is µ(z) > 1, ga > 0) andtherefore if the identified tension has physical origin thiscan not be attributed to an f(R) MG gravity theoryfor any expansion background. Similarly minimal scalartensor theories [3, 122], Horndeski theories [123, 124]and beyond Horndeski Gleyzes-Langlois-Piazza-Vernizzi(GLPV) theories [125] can only produce weaker gravityat low z under very specific and in some cases unnat-ural conditions[52]. For example minimal scalar-tensortheories would require the existence of a phantom com-sological background expansion (equation of state param-eter w < −1)[3, 122]. In fact, a very large class of MG

12

Best fit

fσ8 data

w=-1

Planck ΛCDM

0.0 0.1 0.2 0.3 0.4 0.5 0.60.6

0.7

0.8

0.9

1.0

1.1

Ω0m

σ 8

Best fit

fσ8 data

w to best fit value

Planck ΛCDM

0.0 0.1 0.2 0.3 0.4 0.5 0.60.6

0.7

0.8

0.9

1.0

1.1

Ω0m

σ 8

Best fit

fσ8 data

marginalization over w

Planck ΛCDM

0.0 0.1 0.2 0.3 0.4 0.5 0.60.6

0.7

0.8

0.9

1.0

1.1

Ω0m

σ 8

Best fit

fσ8 - Eg data

w=-1

Planck ΛCDM

0.0 0.1 0.2 0.3 0.4 0.5 0.60.6

0.7

0.8

0.9

1.0

1.1

Ω0m

σ 8

Best fit

fσ8 - Eg data

w to best fit value

Planck ΛCDM

0.0 0.1 0.2 0.3 0.4 0.5 0.60.6

0.7

0.8

0.9

1.0

1.1

Ω0m

σ 8

Best fit

fσ8-Eg data

marginalization over w

Planck ΛCDM

0.0 0.1 0.2 0.3 0.4 0.5 0.60.6

0.7

0.8

0.9

1.0

1.1

Ω0m

σ 8FIG. 9. The confidence contours of the parameter space (σ8-Ωm) in the context of GR (left panels) and in the presence ofthe w parameter (fixing ga = 0 and gb = 0). We have considered both the case of a marginalized w ([−1.5,−0.5]) parametervalue (right panels) and the case of setting w to its best fit value (−0.94 and −1.29 from fσ8(z) and fσ8(z) + EG(z) datarespectively) (middle panels). The red and green dots describe the Planck18/ΛCDM best fit and the best-fit values from data.The EG(z) and fσ8(z) data compilations of datapoints with less correlation from Tables VII and VI of the Appendix B wasused. Notice that the reduction of tension between the best fit parameter values and Planck/ΛCDM is less efficient when thew degree of freedom (modified background expansion rate) is introduced compared to the MG degree of freedom ga shown inFig. 8.

0.15<z<0.43 Amon (2017)

z=0.32 Reyes (2010)

0.15<z<0.43 Blake (2015)

z=0.27 Singh (2018)

– – – GR-ΛCDM Planck18 (z=0.3)

· - · Best Fit (ga=-0.129, gb=-2.308, n=2, m=2, z=0.3)

1 5 10 50 100-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

R [Mpc/h]

EG

0.43<z<0.7 Amon (2017)

z=0.57 Alam (2016)

0.5<z<0.7 de la Torre (2016)

0.7<z<1.2 de la Torre (2016)

0.43<z<0.7 Blake (2015)

z=0.57 Jullo (2019)

z=0.57 Singh (2018)

· · · GR-ΛCDM Planck18 (z=0.7)

——— Best Fit (ga=-0.129, gb=-2.308 , n=2, m=2, z=0.7)

1 2 5 10 20 50-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

R [Mpc/h]

EG

FIG. 10. Measurements of EG as a function of scale R in the range 0.15 < z < 0.43 (left panel) and 0.43 < z < 1.2 (right panel).The data EG(R) from Tables VIII and IX of the Appendix B was used. The dashed black line shows the Planck18/ΛCDMprediction at z = 0.3, the dotted black line shows the Planck18/ΛCDM prediction at z = 0.7, while the dotdashed black lineand the large dashed black line shows the best fit of the EG in the context of parametrizations Eqs.(2.18) and (2.19) at z = 0.3and at z = 0.7 respectively.

13

Param. Planck18/ΛCDM Dataset Dataset Dataset Datasets Datasetsfσ8(z). fσ8(z) EG(z) fσ8(z) + EG(z) fσ8(z) + EG(z)

corr. no corr. corr. no corr.

Ω0m 0.3153± 0.0073 0.289± 0.032 0.283± 0.028 0.285± 0.044 0.288± 0.026 0.282± 0.023σ8 0.8111± 0.0060 0.807± 0.024 0.819± 0.025 0.795± 0.024 0.810± 0.024ga 0 −0.767± 0.299 −0.826± 0.293 −0.621± 0.914 −0.627± 0.291 −0.723± 0.281gb 0 −3.510± 0.605 −3.562± 0.601 −3.563± 0.601

TABLE IV. Planck18/ΛCDM based on TT,TE,EE+lowE+ lensing likelihoods best fit [15] and the best-fit values from datacompilation of datapoints with less correlation.

models, the scalar-tensor Horndeski models, are not con-sistent with the observational indications of weakeninggravity. In fact as stated in Ref. [126] (p. 12), µ for sta-ble Horndeski models is always larger than, or equal to,1 so that matter perturbations in viable Horndeski mod-els always grow faster than the corresponding GR mod-els with the same backgrounds. Thus these MG models(which include f(R) gravities) are unable to account forthe weakening and would provide a worse fit than GR tothe fσ8/EG data. The search for MG models that can ac-count for the observed indications for weakening gravityis thus an interesting extension of the present analysis.

A partial cause of the EG data tension withPlanck/ΛCDM is lensing magnification. As shownin [127, 128] the effects of lensing magnification modifythe galaxy-galaxy lensing correlations as well as galaxy-galaxy correlations and as a consequence introducesystematic errors in the estimate of EG while makingit bias dependent. The effect is small for redshiftssmaller than 1 (about 5 − 10%) but it can become aslarge as 20 − 40% for redshifts z ' 1.5. Thus, thissystematic contribution can be relevant already forDark Energy Survey (DES) [38, 129–132] and certainlyfor higher redshift surveys. However, the magnitudeof lensing contribution at the redshifts of the datacompilation we are using (z < 1) is not large enough tosignificantly reduce the identified tension which existseven at the level of the RSD data alone. The systematiceffect discussed in [127, 128] is important especially forupcoming surveys like Euclid [133] which probe higherredshifts even though even in that case it may not belarge enough to be the only source the observed tension.An interesting feature of our compilation is the scaledependence the EG(R, z) data. This may be used toprobe the parameters of scale dependent MG µ and Σparametrizations which are well motivated physically.We plan to present such constraints elsewhere usingupcoming and more extensive data able to constrain therequired larger parameter space that appears in scaledependent µ and Σ parametrizations. A key questionto address is whether the addition of scale dependencein the parametrizations can improve significantly the

overall fit. No such indications are currently known[120] but this may well change using more extensive andaccurate scale dependent EG and fσ8 data.The introduction of the MG parameters µ and Σ alongwith the variation of the parameters Ωm and σ8 leadsto a model (MG-ΛCDM) that is a much better fit tothe growth fσ8 and EG data than the Planck/ΛCDMmodel in the context of GR. We have called this effecta ’tension’ between the new best fit parameter values(MG-ΛCDM) and the GR-Planck/ΛCDM parametervalues (from Planck18 fit) which are 5 − 6σ away fromthe new best fit parameter values. On the other hand,the MG parameters do not seem to change significantlythe fit of the Planck data as indicated in Ref. [4] and inPlanck18 [15] which indicate that pure CMB data appearto favor GR. Thus, the particular parametrization wehave used does not seem to significantly reduce thetension between CMB and growth/weak lensing datasince MG gravity appears to be favored by growth/weaklensing but not by the CMB. This is an issue we plan toinvestigate in more detail in the future by consideringe.g. different MG parametrizations for the evolution ofthe µ and Σ parameters that will not only improve thefit to the fσ8 /EG data but also improve the fit to theCMB data where some tensions are already evident (e.g.the lensing anomaly discussed in Planck18 [15]).

Supplemental Material: The Mathematica fileused for the numerical analysis and for construction ofthe figures can be found in [134].

ACKNOWLEDGEMENTS

We thank Savvas Nesseris and Ruth Durrer for usefulcomments. This article has benefited from COST ActionCA15117 (CANTATA), supported by COST (EuropeanCooperation in Science and Technology).

14

Space 2D Projected SpaceDataset (Ω0m, σ8, ga) (Ω0m, ga, gb) (Ω0m, σ8, ga, gb) (Ω0m, σ8) (Ω0m, ga) (σ8, ga) (ga, gb) (Ω0m, gb) (σ8, gb)

fσ8(z) corr. 2.39σ 0.22σ 2.19σ 1.79σfσ8(z) no corr. 2.55σ 0.19σ 2.19σ 1.48σ

EG(z) 5.06 0.01σ 4.04σ 6.28σEG(z)+fσ8(z) corr. 5.69 0.36σ 1.35σ 1.59σ 4.31σ 6.12σ 5.12σEG(z)+fσ8(z)no corr. 5.78 0.31σ 2.21σ 1.33σ 4.54σ 6.38σ 5.29σ

TABLE V. Sigma differences of the best fit contours from Planck18/ΛCDM. The EG(z) and fσ8(z) data compilations ofdatapoints with less correlation from Tables VII and VI of the Appendix B was used.

Appendix A: ANALYSIS OF SUBSETS OFDATAPOINTS WITH LESS CORRELATION

In this Appendix we present the results of the statisti-cal analysis of the fσ8(z) and EG(z) data compilationsof datapoints with less correlation. These subsets ofthe data are indicated with bold font in the index ofthe Tables VI and VII of the Appendix B. Using thesesubsets of the data and repeating our analysis we obtainthe best fit parameter values and the tension levelsin both the 2D projections and in the full 3D-4D pa-rameter spaces as shown in Tables IV and V respectively.

These results indicate that even though the tension

level for the combined (EG+fσ8 ) reduces somewhat(from 6σ to about 5.5σ) it remains high enough to causeconcerns for the self consistency of the Planck/ΛCDMmodel and indications for the presence of weakeninggravity.

Appendix B: DATA USED IN THE ANALYSIS

In this appendix we present the data used in theanalysis.

TABLE VI: The fσ8 updated data compilation of Ref. [2] used in thepresent analysis. The subset of the datapoints with less correlation isindicated with bold font in the index.

Index Dataset z fσ8(z) Refs. Year Fiducial Cosmology

1 SDSS-LRG 0.35 0.440± 0.050 [135] 30 October 2006 (Ωm,ΩK , σ8)= (0.25, 0, 0.756)[136]2 VVDS 0.77 0.490± 0.18 [135] 6 October 2009 (Ωm,ΩK , σ8) = (0.25, 0, 0.78)3 2dFGRS 0.17 0.510± 0.060 [135] 6 October 2009 (Ωm,ΩK) = (0.3, 0, 0.9)4 2MRS 0.02 0.314± 0.048 [137], [138] 13 November 2010 (Ωm,ΩK , σ8) = (0.266, 0, 0.65)5 SnIa+IRAS 0.02 0.398± 0.065 [139], [138] 20 October 2011 (Ωm,ΩK , σ8) = (0.3, 0, 0.814)6 SDSS-LRG-200 0.25 0.3512± 0.0583 [140] 9 December 2011 (Ωm,ΩK , σ8) = (0.276, 0, 0.8)7 SDSS-LRG-200 0.37 0.4602± 0.0378 [140] 9 December 20118 SDSS-LRG-60 0.25 0.3665± 0.0601 [140] 9 December 2011 (Ωm,ΩK , σ8) = (0.276, 0, 0.8)9 SDSS-LRG-60 0.37 0.4031± 0.0586 [140] 9 December 2011

10 WiggleZ 0.44 0.413± 0.080 [141] 12 June 2012 (Ωm, h, σ8) = (0.27, 0.71, 0.8)11 WiggleZ 0.60 0.390± 0.063 [141] 12 June 201212 WiggleZ 0.73 0.437± 0.072 [141] 12 June 201213 6dFGS 0.067 0.423± 0.055 [84] 4 July 2012 (Ωm,ΩK , σ8) = (0.27, 0, 0.76)14 SDSS-BOSS 0.30 0.407± 0.055 [142] 11 August 2012 (Ωm,ΩK , σ8) = (0.25, 0, 0.804)15 SDSS-BOSS 0.40 0.419± 0.041 [142] 11 August 201216 SDSS-BOSS 0.50 0.427± 0.043 [142] 11 August 201217 SDSS-BOSS 0.60 0.433± 0.067 [142] 11 August 201218 VIPERS 0.80 0.470± 0.080 [143] 9 July 2013 (Ωm,ΩK , σ8) = (0.25, 0, 0.82)19 SDSS-DR7-LRG 0.35 0.429± 0.089 [144] 8 August 2013 (Ωm,ΩK , σ8)= (0.25, 0, 0.809)[145]20 GAMA 0.18 0.360± 0.090 [146] 22 September 2013 (Ωm,ΩK , σ8) = (0.27, 0, 0.8)21 GAMA 0.38 0.440± 0.060 [146] 22 September 201322 BOSS-LOWZ 0.32 0.384± 0.095 [147] 17 December 2013 (Ωm,ΩK , σ8) = (0.274, 0, 0.8)23 SDSS DR10 and DR11 0.32 0.48± 0.10 [147] 17 December 2013 (Ωm,ΩK , σ8)= (0.274, 0, 0.8)[148]

15

24 SDSS DR10 and DR11 0.57 0.417± 0.045 [147] 17 December 201325 SDSS-MGS 0.15 0.490± 0.145 [149] 30 January 2015 (Ωm, h, σ8) = (0.31, 0.67, 0.83)26 SDSS-veloc 0.10 0.370± 0.130 [150] 16 June 2015 (Ωm,ΩK , σ8)= (0.3, 0, 0.89)[151]27 FastSound 1.40 0.482± 0.116 [152] 25 November 2015 (Ωm,ΩK , σ8)= (0.27, 0, 0.82)[153]28 SDSS-CMASS 0.59 0.488± 0.060 [154] 8 July 2016 (Ωm, h, σ8) = (0.307115, 0.6777, 0.8288)29 BOSS DR12 0.38 0.497± 0.045 [155] 11 July 2016 (Ωm,ΩK , σ8) = (0.31, 0, 0.8)30 BOSS DR12 0.51 0.458± 0.038 [155] 11 July 201631 BOSS DR12 0.61 0.436± 0.034 [155] 11 July 201632 BOSS DR12 0.38 0.477± 0.051 [156] 11 July 2016 (Ωm, h, σ8) = (0.31, 0.676, 0.8)33 BOSS DR12 0.51 0.453± 0.050 [156] 11 July 201634 BOSS DR12 0.61 0.410± 0.044 [156] 11 July 201635 VIPERS v7 0.76 0.440± 0.040 [157] 26 October 2016 (Ωm, σ8) = (0.308, 0.8149)36 VIPERS v7 1.05 0.280± 0.080 [157] 26 October 201637 BOSS LOWZ 0.32 0.427± 0.056 [158] 26 October 2016 (Ωm,ΩK , σ8) = (0.31, 0, 0.8475)38 BOSS CMASS 0.57 0.426± 0.029 [158] 26 October 201639 VIPERS 0.727 0.296± 0.0765 [159] 21 November 2016 (Ωm,ΩK , σ8) = (0.31, 0, 0.7)40 6dFGS+SnIa 0.02 0.428± 0.0465 [160] 29 November 2016 (Ωm, h, σ8) = (0.3, 0.683, 0.8)41 VIPERS PDR2 0.60 0.550± 0.120 [161] 16 December 2016 (Ωm,Ωb, σ8) = (0.3, 0.045, 0.823)42 VIPERS PDR2 0.86 0.400± 0.110 [161] 16 December 201643 SDSS DR13 0.1 0.48± 0.16 [162] 22 December 2016 (Ωm, σ8)= (0.25, 0.89)[151]44 2MTF 0.001 0.505± 0.085 [163] 16 June 2017 (Ωm, σ8) = (0.3121, 0.815)45 VIPERS PDR2 0.85 0.45± 0.11 [164] 31 July 2017 (Ωb,Ωm, h) = (0.045, 0.30, 0.8)46 BOSS DR12 0.31 0.384± 0.083 [165] 15 September 2017 (Ωm, h, σ8) = (0.307, 0.6777, 0.8288)47 BOSS DR12 0.36 0.409± 0.098 [165] 15 September 201748 BOSS DR12 0.40 0.461± 0.086 [165] 15 September 201749 BOSS DR12 0.44 0.426± 0.062 [165] 15 September 201750 BOSS DR12 0.48 0.458± 0.063 [165] 15 September 201751 BOSS DR12 0.52 0.483± 0.075 [165] 15 September 201752 BOSS DR12 0.56 0.472± 0.063 [165] 15 September 201753 BOSS DR12 0.59 0.452± 0.061 [165] 15 September 201754 BOSS DR12 0.64 0.379± 0.054 [165] 15 September 201755 SDSS DR7 0.1 0.376± 0.038 [166] 12 December 2017 (Ωm,Ωb, σ8) = (0.282, 0.046, 0.817)56 SDSS-IV 1.52 0.420± 0.076 [167] 8 January 2018 (Ωm,Ωbh

2, σ8) = (0.26479, 0.02258, 0.8)57 SDSS-IV 1.52 0.396± 0.079 [168] 8 January 2018 (Ωm,Ωbh

2, σ8) = (0.31, 0.022, 0.8225)58 SDSS-IV 0.978 0.379± 0.176 [169] 9 January 2018 (Ωm, σ8) = (0.31, 0.8)59 SDSS-IV 1.23 0.385± 0.099 [169] 9 January 201860 SDSS-IV 1.526 0.342± 0.070 [169] 9 January 201861 SDSS-IV 1.944 0.364± 0.106 [169] 9 January 201862 VIPERS PDR2 0.60 0.49± 0.12 [170] 6 June 2018 (Ωb,Ωm, h, σ8) = (0.045, 0.31, 0.7, 0.8)63 VIPERS PDR2 0.86 0.46± 0.09 [170] 6 June 201864 BOSS DR12 voids 0.57 0.501± 0.051 [171] 1 April 2019 (Ωb,Ωm, h, σ8) = (0.0482, 0.307, 0.6777, 0.8228)65 2MTF 6dFGSv 0.03 0.404± 0.0815 [172] 7 June 2019 (Ωb,Ωm, h, σ8) = (0.0491, 0.3121, 0.6571, 0.815)66 SDSS-IV 0.72 0.454± 0.139 [173] 17 September 2019 (Ωm,Ωbh

2, σ8) = (0.31, 0.022, 0.8)

TABLE VII: The EG(z) data compilation used in the present analysis.The subset of the datapoints with less correlation is indicated with boldfont in the index.

Index Dataset z EG(z) σEG Scale [Mpc/h] Reference

1 KiDS GAMA 0.267 0.43 0.13 5 < R < 40 [174]2 SDSS BOSS LOWZ 0.27 0.40 0.05 25 < R < 150 [175]3 CMB lens BOSS LOWZ 0.27 0.46 0.085 25 < R < 150 [175]4 KiDS 2dFLenS BOSS LOWZ 2dFLOZ 0.305 0.27 0.08 5 < R < 60 [174]5 RCSLenS CFHTLenS WiggleZ BOSS WGZLoZ LOWZ 0.32 0.40 0.09 R > 3 [176]6 RCSLenS CFHTLenS WiggleZ BOSS WGZLoZ LOWZ 0.32 0.48 0.10 R > 10 [176]7 SDSS 0.32 0.39 0.06 10 < Rp < 50 [70]8 KiDS 2dFLenS BOSS CMASS 2dFHIZ 0.554 0.26 0.07 5 < R < 60 [174]9 RCSLenS CFHTLenS WiggleZ BOSS WGZHiZ CMASS 0.57 0.31 0.06 R > 3 [176]10 RCSLenS CFHTLenS WiggleZ BOSS WGZHiZ CMASS 0.57 0.30 0.07 R > 10 [176]11 SDSS-III BOSS CMB lens CMASS 0.57 0.24 0.06 R > 150 [73]12 CFHTLenS SDSS-III BOSS CMASS 0.57 0.42 0.056 5 < R < 26 [177]13 CMB lens BOSS CMASS 0.57 0.39 0.05 25 < R < 150 [175]14 CFHTLenS BOSS CMASS 0.57 0.43 0.10 10 < R < 60 [178]

16

15 CFHTLenS VIPERS 0.60 0.16 0.09 3 < R < 20 [68]16 CFHTLenS VIPERS 0.86 0.09 0.07 3 < R < 20 [68]

TABLE VIII: The EG(R) data compilation in the range 0.15 < z < 0.43used in the present analysis.

Index R[Mpc/h] EG(R) σEG z Reference

1 3.61 0.37 0.10 0.27 [175]2 4.91 0.42 0.08 0.27 [175]3 6.60 0.50 0.07 0.27 [175]4 9.07 0.39 0.07 0.27 [175]5 12.20 0.37 0.06 0.27 [175]6 16.58 0.45 0.06 0.27 [175]7 22.54 0.32 0.04 0.27 [175]8 30.30 0.39 0.05 0.27 [175]9 41.19 0.44 0.06 0.27 [175]10 55.99 0.45 0.08 0.27 [175]11 76.98 0.34 0.10 0.27 [175]12 103.47 0.28 0.15 0.27 [175]13 2.45 0.28 0.23 0.32 [70]14 3.41 0.49 0.16 0.32 [70]15 4.64 0.50 0.12 0.32 [70]16 6.62 0.32 0.09 0.32 [70]17 9.85 0.34 0.07 0.32 [70]18 14.83 0.45 0.08 0.32 [70]19 22.10 0.43 0.09 0.32 [70]20 45.87 0.32 0.10 0.32 [70]21 1.76 0.74 0.21 0.15-0.43 [176]22 2.23 0.71 0.15 0.15-0.43 [176]23 2.85 0.35 0.14 0.15-0.43 [176]24 3.56 0.30 0.11 0.15-0.43 [176]25 4.45 0.35 0.11 0.15-0.43 [176]26 5.65 0.28 0.10 0.15-0.43 [176]27 7.059 0.43 0.11 0.15-0.43 [176]28 8.94 0.45 0.11 0.15-0.43 [176]29 11.33 0.47 0.12 0.15-0.43 [176]30 14.34 0.55 0.12 0.15-0.43 [176]31 17.98 0.40 0.12 0.15-0.43 [176]32 22.21 0.37 0.14 0.15-0.43 [176]33 28.88 0.39 0.18 0.15-0.43 [176]34 36.15 0.35 0.19 0.15-0.43 [176]35 45.26 0.30 0.30 0.15-0.43 [176]36 5.01 0.25 0.16 0.15-0.43 [174]37 5.37 0.39 0.16 0.15-0.43 [174]38 5.58 0.094 0.18 0.15-0.43 [174]39 8.15 0.30 0.14 0.15-0.43 [174]40 8.57 0.41 0.14 0.15-0.43 [174]41 9.02 0.41 0.24 0.15-0.43 [174]42 13.23 0.49 0.16 0.15-0.43 [174]43 13.95 0.43 0.16 0.15-0.43 [174]44 14.76 0.15 0.17 0.15-0.43 [174]45 21.08 0.51 0.23 0.15-0.43 [174]46 22.75 0.33 0.23 0.15-0.43 [174]47 23.96 0.32 0.32 0.15-0.43 [174]48 35.52 0.33 0.29 0.15-0.43 [174]49 36.98 0.40 0.33 0.15-0.43 [174]50 39.00 0.32 0.38 0.15-0.43 [174]51 56.60 0.37 0.80 0.15-0.43 [174]

TABLE IX: The EG(R) data compilation in the range 0.43 < z < 1.2used in the present analysis.

Index R[Mpc/h] EG(R) σEG z Reference

17

1 5.13 0.23 0.14 0.43-0.7 [174]2 5.69 0.19 0.19 0.43-0.7 [174]3 8.28 0.32 0.12 0.43-0.7 [174]4 9.19 0.27 0.17 0.43-0.7 [174]5 13.69 0.21 0.12 0.43-0.7 [174]6 14.98 0.46 0.25 0.43-0.7 [174]7 24.43 0.95 0.47 0.43-0.7 [174]8 22.02 0.22 0.13 0.43-0.7 [174]9 36.28 0.48 0.18 0.43-0.7 [174]10 39.84 0.84 0.57 0.43-0.7 [174]11 59.78 0.54 0.45 0.43-0.7 [174]12 1.74 0.34 0.29 0.43-0.7 [176]13 2.25 0.31 0.17 0.43-0.7 [176]14 2.74 0.57 0.14 0.43-0.7 [176]15 3.46 0.43 0.11 0.43-0.7 [176]16 4.45 0.35 0.10 0.43-0.7 [176]17 5.56 0.30 0.09 0.43-0.7 [176]18 6.92 0.24 0.09 0.43-0.7 [176]19 8.84 0.28 0.08 0.43-0.7 [176]20 11.15 0.26 0.08 0.43-0.7 [176]21 13.90 0.29 0.07 0.43-0.7 [176]22 17.74 0.24 0.08 0.43-0.7 [176]23 21.96 0.25 0.09 0.43-0.7 [176]24 27.80 0.32 0.10 0.43-0.7 [176]25 34.81 0.41 0.14 0.43-0.7 [176]26 44.26 0.62 0.19 0.43-0.7 [176]27 2.17 0.38 0.15 0.57 [177]28 3.30 0.26 0.11 0.57 [177]29 4.99 0.46 0.09 0.57 [177]30 7.56 0.42 0.08 0.57 [177]31 11.38 0.35 0.08 0.57 [177]32 17.26 0.43 0.09 0.57 [177]33 25.99 0.43 0.11 0.57 [177]34 39.60 0.33 0.15 0.57 [177]35 2.56 0.48 0.08 0.57 [175]36 3.40 0.36 0.09 0.57 [175]37 4.60 0.32 0.08 0.57 [175]38 6.12 0.40 0.08 0.57 [175]39 8.20 0.33 0.07 0.57 [175]40 11.00 0.43 0.09 0.57 [175]41 14.87 0.32 0.07 0.57 [175]42 19.75 0.34 0.12 0.57 [175]43 26.21 0.31 0.15 0.57 [175]44 35.44 0.22 0.18 0.57 [175]45 2.22 0.27 0.17 0.57 [178]46 3.56 0.13 0.23 0.57 [178]47 5.64 0.16 0.19 0.57 [178]48 8.86 0.49 0.28 0.57 [178]49 14.13 0.64 0.22 0.57 [178]50 22.54 0.29 0.14 0.57 [178]51 35.37 0.40 0.21 0.57 [178]52 2.64 0.31 0.14 0.5-0.7 [68]53 4.17 0.20 0.15 0.5-0.7 [68]54 6.63 0.22 0.17 0.5-0.7 [68]55 10.44 0.01 0.20 0.5-0.7 [68]56 16.37 0.09 0.26 0.5-0.7 [68]57 2.61 0.34 0.12 0.7-1.2 [68]58 4.15 0.06 0.12 0.7-1.2 [68]59 6.60 0.11 0.13 0.7-1.2 [68]60 10.42 0.01 0.16 0.7-1.2 [68]61 16.56 0.10 0.21 0.7-1.2 [68]

18

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